## Simple Linear Regression: How It works? (Python Implementation)

 Simple Linear Regression: How It works? (Python Implementation)

# Linear Regression (Python Implementation)

This article discusses the basics of linear regression and its implementation in Python programming language.
Linear regression is a statistical approach for modelling the relationship between a dependent variable with a given set of independent variables.
Note: In this article, we refer dependent variables as response and independent variables as features for simplicity.
In order to provide a basic understanding of linear regression, we start with the most basic version of linear regression, i.e. Simple linear regression.

## Simple Linear Regression

Simple linear regression is an approach for predicting a response using a single feature.

It is assumed that the two variables are linearly related. Hence, we try to find a linear function that predicts the response value(y) as accurately as possible as a function of the feature or independent variable(x).
Let us consider a dataset where we have a value of response y for every feature x:
For generality, we define:
x as feature vector, i.e x = [x_1, x_2, …., x_n],
y as response vector, i.e y = [y_1, y_2, …., y_n]
for n observations (in above example, n=10).
A scatter plot of above dataset looks like:-
Now, the task is to find a line which fits best in above scatter plot so that we can predict the response for any new feature values. (i.e a value of x not present in the dataset)

This line is called the regression line.
The equation of the regression line is represented as:
Here,
• h(x_i) represents the predicted response value for ith observation.
• b_0 and b_1 are regression coefficients and represent y-intercept and slope of regression line respectively.
To create our model, we must “learn” or estimate the values of regression coefficients b_0 and b_1. And once we’ve estimated these coefficients, we can use the model to predict responses!
In this article, we are going to use the Least Squares technique.
Now consider:
Here, e_i is a residual error in ith observation.
So, our aim is to minimize the total residual error.
We define the squared error or cost function, J as:

and our task is to find the value of b_0 and b_1 for which J(b_0,b_1) is minimum!
Without going into the mathematical details, we present the result here:
where SS_xy is the sum of cross-deviations of y and x:
and SS_xx is the sum of squared deviations of x:
Note: The complete derivation for finding least squares estimates in simple linear regression can be found here.
Given below is the python implementation of the above technique on our small dataset:
 import numpy as np  import matplotlib.pyplot as plt     def estimate_coef(x, y):      # number of observations/points      n = np.size(x)         # mean of x and y vector      m_x, m_y = np.mean(x), np.mean(y)         # calculating cross-deviation and deviation about x      SS_xy = np.sum(y*x) - n*m_y*m_x      SS_xx = np.sum(x*x) - n*m_x*m_x         # calculating regression coefficients      b_1 = SS_xy / SS_xx      b_0 = m_y - b_1*m_x         return(b_0, b_1)     def plot_regression_line(x, y, b):      # plotting the actual points as scatter plot      plt.scatter(x, y, color = "m",                 marker = "o", s = 30)         # predicted response vector      y_pred = b[0] + b[1]*x         # plotting the regression line      plt.plot(x, y_pred, color = "g")         # putting labels      plt.xlabel('x')      plt.ylabel('y')         # function to show plot      plt.show()     def main():      # observations      x = np.array([0, 1, 2, 3, 4, 5, 6, 7, 8, 9])      y = np.array([1, 3, 2, 5, 7, 8, 8, 9, 10, 12])         # estimating coefficients      b = estimate_coef(x, y)      print("Estimated coefficients:\nb_0 = {}  \            \nb_1 = {}".format(b[0], b[1]))         # plotting regression line      plot_regression_line(x, y, b)     if __name__ == "__main__":      main()
The output of the above piece of code is:
Estimated coefficients:
b_0 = -0.0586206896552
b_1 = 1.45747126437

And the graph obtained looks like this:

## http://bit.ly/2Ufe34U

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1. for your estimated coefficients I seem to be getting different figures
b_0 = 1.2363636363636363
b_1 = 1.1696969696969697

1. It will vary with the system configuration

2. Ok I get you, I further used excel linear regression plotting and got the same result like I got in python